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%I #13 Dec 13 2022 07:56:03
%S 1,1,1,2,3,5,6,10,12,19,22,34,40,60,69,101,118,168,195,272,317,434,
%T 505,679,793,1050,1224,1599,1867,2409,2811,3587,4186,5290,6168,7724,
%U 9005,11186,13026,16062,18692,22894,26613,32394,37619,45535,52815,63593,73680
%N Number of integer partitions of n with reverse-alternating product <= 1.
%C Includes all partitions of even length (A027187).
%C Also the number of integer partitions of n with reverse-alternating sum <= 1.
%C Also the number of integer partitions of n having either even length (A027187) or having exactly one odd part in the conjugate partition (A100824).
%C We define the alternating product of a sequence (y_1,...,y_k) to be Product_i y_i^((-1)^(i-1)). The reverse-alternating product is the alternating product of the reversed sequence.
%F a(n) = A027187(n) + A035363(n-1) for n >= 1. [Corrected by _Georg Fischer_, Dec 13 2022]
%F a(n) = A119620(n) + A344608(n).
%e The a(1) = 1 through a(8) = 12 partitions:
%e (1) (11) (21) (22) (32) (33) (43) (44)
%e (111) (31) (41) (42) (52) (53)
%e (1111) (221) (51) (61) (62)
%e (2111) (2211) (331) (71)
%e (11111) (3111) (2221) (2222)
%e (111111) (3211) (3221)
%e (4111) (3311)
%e (22111) (4211)
%e (211111) (5111)
%e (1111111) (221111)
%e (311111)
%e (11111111)
%t altprod[q_]:=Product[q[[i]]^(-1)^(i-1),{i,Length[q]}];
%t Table[Length[Select[IntegerPartitions[n],altprod[Reverse[#]]<=1&]],{n,0,30}]
%Y The odd-length case is A035363 (shifted).
%Y The strict case is A067661.
%Y The non-reverse version is counted by A119620, ranked by A347466.
%Y The even bisection is A236913.
%Y The opposite version (>= instead of <=) is A344607.
%Y The case of < 1 instead of <= 1 is A344608.
%Y The multiplicative version (factorizations) is A347438, non-reverse A339846.
%Y Allowing any integer reverse-alternating product gives A347445.
%Y The complement (> 1 instead of <= 1) is counted by A347449.
%Y Ranked by A347465, non-reverse A347450.
%Y A000041 counts partitions.
%Y A027187 counts partitions of even length.
%Y A027193 counts partitions of odd length.
%Y A058622 counts compositions with alternating sum <= 0 (A294175 for < 0).
%Y A100824 counts partitions with alternating sum <= 1.
%Y A103919 counts partitions by sum and alternating sum (reverse: A344612).
%Y A347461 counts possible alternating products of partitions.
%Y A347462 counts possible reverse-alternating products of partitions.
%Y Cf. A000070, A038548, A086543, A116406, A325534, A325535, A344611, A344654, A344740, A347440, A347442, A347446, A347448.
%K nonn
%O 0,4
%A _Gus Wiseman_, Sep 14 2021